How does a mathematical proof show that a theorem follows from axioms?

In mathematics, a proof is a rigorous argument that establishes the validity of a proposition. Within a formal system, a proof is structured as a finite sequence of well-formed formulas that demonstrates how a theorem follows from a set of axioms.

This process functions according to the following principles:

• Foundational Basis: Each formula in the sequence must either be an axiom—a starting assumption—or be derived from preceding formulas in the sequence.12
• Rules of Inference: The transition from one formula to the next is governed by established rules of inference. These rules ensure that the logical connection between the premises and the conclusion is sound.13
• Completion: The sequence is considered a valid proof when the final formula in the chain is the specific statement intended to be proved.12

By adhering to these strict logical steps, mathematicians ensure that theorems are not merely intuitive, but are necessary consequences of the underlying axiomatic framework.32